American Institute of Mathematical Sciences

Advanced
2015, 12(1): 83-97. doi: 10.3934/mbe.2015.12.83

Delayed population models with Allee effects and exploitation

Eduardo Liz 1, and  Alfonso Ruiz-Herrera 2,

1. 

Departamento de Matemática Aplicada II, E.T.S.E. Telecomunicación, Universidade de Vigo, Campus Marcosende, 36310 Vigo
2. 

Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., H-6720 Szeged, Hungary

Received  April 2014 Revised  October 2014 Published  December 2014

Allee effects make populations more vulnerable to extinction, especially under severe harvesting or predation. Using a delay-differential equation modeling the evolution of a single-species population subject to constant effort harvesting, we show that the interplay between harvest strength and Allee effects leads not only to collapses due to overexploitation; large delays can interact with Allee effects to produce extinction at population densities that would survive for smaller time delays. In case of bistability, our estimations on the basins of attraction of the two coexisting attractors improve some recent results in this direction. Moreover, we show that the persistent attractor can exhibit bubbling: a stable equilibrium loses its stability as harvesting effort increases, giving rise to sustained oscillations, but higher mortality rates stabilize the equilibrium again.
Keywords: bifurcation., bistability, difference equation, global stability, delay differential equation, Allee effect, Population model.
Mathematics Subject Classification: Primary: 34K18, 34K20, 92D25; Secondary: 37E0.
Citation: Eduardo Liz, Alfonso Ruiz-Herrera. Delayed population models with Allee effects and exploitation. Mathematical Biosciences & Engineering, 2015, 12 (1) : 83-97. doi: 10.3934/mbe.2015.12.83
References:
[1]

D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters, J. Theoret. Biol., 218 (2002), 375-394. doi: 10.1006/jtbi.2002.3084.

[2]

B. Cid, F. M. Hilker and E. Liz, Harvest timing and its population dynamic consequences in a discrete single-species model, Math. Biosci., 248 (2014), 78-87. doi: 10.1016/j.mbs.2013.12.003.

[3]

C. W. Clark, Mathematical Bioeconomics. Optimal Management of Renewable Resources, $2^{nd}$ edition, John Wiley & Sons, Hoboken, New Jersey, 1990.

[4]

F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, New York, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001.

[5]

A. M. De Roos and L. Persson, Size-dependent life-history traits promote catastrophic collapses of top predators, Proc. Natl. Acad. Sci. USA, 99 (2002), 12907-12912.

[6]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[7]

S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, J. Biol. Dyn., 4 (2010), 397-408. doi: 10.1080/17513750903377434.

[8]

S. A. H. Geritz and E. Kisdi, Mathematical ecology: Why mechanistic models?, J. Math. Biol., 65 (2012), 1411-1415. doi: 10.1007/s00285-011-0496-3.

[9]

K. P. Hadeler, Neutral delay equations from and for population dynamics, Electron. J. Qual. Theory Differ. Equ., 11 (2008), 1-18.

[10]

C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations, 256 (2014), 2101-2114. doi: 10.1016/j.jde.2013.12.015.

[11]

A. F. Ivanov, E. Liz and S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations, Differential Equations Dynam. Systems, 11 (2003), 33-54.

[12]

A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, Dynam. Report. Expositions Dynam. Systems (N.S.), 1 (1992), 164-224.

[13]

M. Jankovic and S. Petrovskii, Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect, Theor. Ecol., 7 (2014), 335-349. doi: 10.1007/s12080-014-0222-z.

[14]

T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hungar., 56 (2008), 83-95. doi: 10.1007/s10998-008-5083-x.

[15]

T. Krisztin and E. Liz, Bubbles for a class of delay differential equations, Qual. Theory Dyn. Syst., 10 (2011), 169-196. doi: 10.1007/s12346-011-0055-8.

[16]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.

[17]

E. Liz, Complex dynamics of survival and extinction in simple population models with harvesting, Theor. Ecol., 3 (2010), 209-221. doi: 10.1007/s12080-009-0064-2.

[18]

E. Liz, M. Pinto, V. Tkachenko and S. Trofimchuk, A global stability criterion for a family of delayed population models, Quart. Appl. Math., 63 (2005), 56-70.

[19]

E. Liz and G. Röst, On the global attractor of delay differential equations with unimodal feedback, Discrete Contin. Dyn. Syst., 24 (2009), 1215-1224. doi: 10.3934/dcds.2009.24.1215.

[20]

E. Liz and A. Ruiz-Herrera, Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback, J. Differential Equations, 255 (2013), 4244-4266. doi: 10.1016/j.jde.2013.08.007.

[21]

E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal., 35 (2003), 596-622. doi: 10.1137/S0036141001399222.

[22]

J. Mallet-Paret and R. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl., 145 (1986), 33-128. doi: 10.1007/BF01790539.

[23]

G. Röst, On the global attractivity controversy for a delay model of hematopoiesis, Appl. Math. Comput., 190 (2007), 846-850. doi: 10.1016/j.amc.2007.01.103.

[24]

G. Röst and J. Wu, Domain decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669. doi: 10.1098/rspa.2007.1890.

[25]

S. Ruan, Delay differential equations in single species dynamics, in Delay differential equations and applications, NATO Sci. Ser. II Math. Phys. Chem., Springer, Dordrecht, 205 (2006), 477-517. doi: 10.1007/1-4020-3647-7_11.

[26]

S. J. Schreiber, Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260. doi: 10.1007/s002850000070.

[27]

S. J. Schreiber, Allee effect, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209. doi: 10.1016/S0040-5809(03)00072-8.

[28]

A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps, Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-94-015-8897-3.

[29]

A. N. Sharkovsky, Y. L. Maistrenko and E. Y. Romanenko, Difference Equations and Their Applications, Kluwer Academic Publishers, Dordrecht, 1993. doi: 10.1007/978-94-011-1763-0.

[30]

D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267. doi: 10.1137/0135020.

[31]

A.-A. Yakubu, N. Li, J. M. Conrad and M.-L. Zeeman, Constant proportion harvest policies: Dynamic implications in the Pacific halibut and Atlantic cod fisheries, Math. Biosci., 232 (2011), 66-77. doi: 10.1016/j.mbs.2011.04.004.

[32]

T. Yi and X. Zou, Maps dynamics versus dynamics of associated delay reaction-diffusion equation with a Neumann condition, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973. doi: 10.1098/rspa.2009.0650.

show all references

References:
[1]

D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters, J. Theoret. Biol., 218 (2002), 375-394. doi: 10.1006/jtbi.2002.3084.

[2]

B. Cid, F. M. Hilker and E. Liz, Harvest timing and its population dynamic consequences in a discrete single-species model, Math. Biosci., 248 (2014), 78-87. doi: 10.1016/j.mbs.2013.12.003.

[3]

C. W. Clark, Mathematical Bioeconomics. Optimal Management of Renewable Resources, $2^{nd}$ edition, John Wiley & Sons, Hoboken, New Jersey, 1990.

[4]

F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, New York, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001.

[5]

A. M. De Roos and L. Persson, Size-dependent life-history traits promote catastrophic collapses of top predators, Proc. Natl. Acad. Sci. USA, 99 (2002), 12907-12912.

[6]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[7]

S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, J. Biol. Dyn., 4 (2010), 397-408. doi: 10.1080/17513750903377434.

[8]

S. A. H. Geritz and E. Kisdi, Mathematical ecology: Why mechanistic models?, J. Math. Biol., 65 (2012), 1411-1415. doi: 10.1007/s00285-011-0496-3.

[9]

K. P. Hadeler, Neutral delay equations from and for population dynamics, Electron. J. Qual. Theory Differ. Equ., 11 (2008), 1-18.

[10]

C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations, 256 (2014), 2101-2114. doi: 10.1016/j.jde.2013.12.015.

[11]

A. F. Ivanov, E. Liz and S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations, Differential Equations Dynam. Systems, 11 (2003), 33-54.

[12]

A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, Dynam. Report. Expositions Dynam. Systems (N.S.), 1 (1992), 164-224.

[13]

M. Jankovic and S. Petrovskii, Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect, Theor. Ecol., 7 (2014), 335-349. doi: 10.1007/s12080-014-0222-z.

[14]

T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hungar., 56 (2008), 83-95. doi: 10.1007/s10998-008-5083-x.

[15]

T. Krisztin and E. Liz, Bubbles for a class of delay differential equations, Qual. Theory Dyn. Syst., 10 (2011), 169-196. doi: 10.1007/s12346-011-0055-8.

[16]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.

[17]

E. Liz, Complex dynamics of survival and extinction in simple population models with harvesting, Theor. Ecol., 3 (2010), 209-221. doi: 10.1007/s12080-009-0064-2.

[18]

E. Liz, M. Pinto, V. Tkachenko and S. Trofimchuk, A global stability criterion for a family of delayed population models, Quart. Appl. Math., 63 (2005), 56-70.

[19]

E. Liz and G. Röst, On the global attractor of delay differential equations with unimodal feedback, Discrete Contin. Dyn. Syst., 24 (2009), 1215-1224. doi: 10.3934/dcds.2009.24.1215.

[20]

E. Liz and A. Ruiz-Herrera, Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback, J. Differential Equations, 255 (2013), 4244-4266. doi: 10.1016/j.jde.2013.08.007.

[21]

E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal., 35 (2003), 596-622. doi: 10.1137/S0036141001399222.

[22]

J. Mallet-Paret and R. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl., 145 (1986), 33-128. doi: 10.1007/BF01790539.

[23]

G. Röst, On the global attractivity controversy for a delay model of hematopoiesis, Appl. Math. Comput., 190 (2007), 846-850. doi: 10.1016/j.amc.2007.01.103.

[24]

G. Röst and J. Wu, Domain decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669. doi: 10.1098/rspa.2007.1890.

[25]

S. Ruan, Delay differential equations in single species dynamics, in Delay differential equations and applications, NATO Sci. Ser. II Math. Phys. Chem., Springer, Dordrecht, 205 (2006), 477-517. doi: 10.1007/1-4020-3647-7_11.

[26]

S. J. Schreiber, Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260. doi: 10.1007/s002850000070.

[27]

S. J. Schreiber, Allee effect, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209. doi: 10.1016/S0040-5809(03)00072-8.

[28]

A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps, Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-94-015-8897-3.

[29]

A. N. Sharkovsky, Y. L. Maistrenko and E. Y. Romanenko, Difference Equations and Their Applications, Kluwer Academic Publishers, Dordrecht, 1993. doi: 10.1007/978-94-011-1763-0.

[30]

D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267. doi: 10.1137/0135020.

[31]

A.-A. Yakubu, N. Li, J. M. Conrad and M.-L. Zeeman, Constant proportion harvest policies: Dynamic implications in the Pacific halibut and Atlantic cod fisheries, Math. Biosci., 232 (2011), 66-77. doi: 10.1016/j.mbs.2011.04.004.

[32]

T. Yi and X. Zou, Maps dynamics versus dynamics of associated delay reaction-diffusion equation with a Neumann condition, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973. doi: 10.1098/rspa.2009.0650.

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